We show the calculation results of parameters \(\alpha \)  and \(\beta /H_{\ast }\)  in the left plot of Fig. REF , the colored bar indicates the nucleation temperature \(T_n\) . The distribution of \(T_n\)  are given in the right plot of Fig. REF .
We find that 8564 out of 18047 sample points satisfy the bubble nucleation condition in Eq. (REF ).
Note that in the calculations of \(\beta /H_{\ast }\)  with Eq. (REF ), we have assumed that reheating
is not significant for typical transitions. In this case, the temperature for GW generation \(T_{\ast }\)  is approximately equivalent to the nucleation temperature \(T_n\) , i.e., \(T_{\ast }\simeq T_{n}\) .
A strong supercooling could not only enhance the strength of the phase transition, but also change the evolution of the Universe since the expansion of the Universe would be dominated by vacuum energy in the supercooled phase, instead of radiation.
In this case, there is a lower bound on the temperature of the transition to ensure the successful bubble percolation and completion of the EWPT [1]}.
